Even as the special counsel, Jack Smith, appears to be edging closer toward bringing charges against former President Donald J. Trump in connection with his efforts to overturn the 2020 election, prosecutors have been continuing to investigate multiple strands of the case.

In recent weeks, Mr. Smith’s team has pushed forward in collecting new evidence and in arranging new interviews with witnesses who could shed light on Mr. Trump’s mind-set in the chaotic postelection period or on other subjects important to the inquiry. At the same time, word has emerged of previously undisclosed investigative efforts, hinting at the breadth and scope of the issues prosecutors are examining.

In the past few days, a lawyer for Bernard B. Kerik, the former New York City police commissioner who worked closely after the election with Mr. Trump’s lawyer, Rudolph W. Giuliani, gave hundreds of pages of documents to prosecutors working with Mr. Smith.

The documents detailed efforts by Mr. Kerik and Mr. Giuliani to identify and investigate allegations of fraud in the election — an issue that is likely to be front and center as prosecutors seek to understand what Mr. Trump may have been thinking when he set in motion various efforts to maintain his grip on power.

While it remains unclear precisely when Mr. Smith may seek an indictment of the former president, the clearest signal yet that one was in the offing came last week from Mr. Trump, who announced on social media that he had received a so-called target letter from prosecutors alluding to at least three charges he might face.

Those charges included conspiracy to defraud the United States, obstruction of an official proceeding and a Reconstruction-era civil rights statute that makes it a crime for people to conspire to threaten or intimidate others from exercising rights provided to them by federal law or the Constitution.

It is not uncommon for prosecutors to keep investigating a criminal case up to the moment an indictment is returned. They can even press forward after charges are filed. But prosecutors are not supposed to use a grand jury of the sort that has been used to investigate Mr. Trump to gather fresh evidence after charges are brought — unless they intend to use the information to seek additional charges.

The production of documents by Mr. Kerik, who was convicted of tax fraud but pardoned by Mr. Trump, came even as his lawyer, Timothy Parlatore, was arranging for Mr. Kerik to sit down with Mr. Smith’s prosecutors for a voluntary interview next month. Mr. Giuliani did a similar interview with Mr. Smith’s team in June.

Among the previously unknown steps taken by Mr. Smith’s team was an interview conducted about three months ago with Richard P. Donoghue, a former top official in the Justice Department at the end of Mr. Trump’s time in office. NBC News reported on the interview on Monday night, and Mr. Donoghue confirmed on Tuesday that it took place. But he declined to comment on what he discussed with Mr. Smith’s prosecutors.

In late 2021, Mr. Donoghue, who served as the acting deputy attorney general under Mr. Trump, told the House select committee investigating Jan. 6 that he and Jeffrey Rosen, the acting attorney general at the time, repeatedly sought to rebuff Mr. Trump’s claims that the election had been marred by widespread fraud. At one point, Mr. Donoghue testified, Mr. Trump urged him and Mr. Rosen to “just say the election was corrupt and leave the rest to me and the Republican congressmen.”

Mr. Donoghue also told the committee that in the waning days of his presidency, Mr. Trump wanted to replace Mr. Rosen with Jeffrey Clark, a loyalist within the Justice Department. Mr. Clark, whose home was searched as part of the election interference inquiry into Mr. Trump, had helped to a draft a letter suggesting that fraud had affected the election results and urging Gov. Brian Kemp of Georgia, a Republican, to call for the creation of a fake slate of electors to the Electoral College declaring that Mr. Trump had won that state, not Joseph R. Biden Jr.

Mr. Smith’s team has also reached out to Mr. Kemp seeking an interview, Garrison Douglas, a spokesman for Mr. Kemp, said on Tuesday. But Mr. Douglas declined to say whether the interview, which was reported by The Washington Post, had been merely scheduled or had already taken place.

Georgia was a key location in Mr. Trump’s campaign to pressure local officials to throw him the election in their states. Brad Raffensperger, Georgia’s secretary of state, recorded Mr. Trump on a phone call in early January 2021, asking him to “find” sufficient votes for him to win the state.

Mr. Smith’s prosecutors have also shown interest in a different line of inquiry in recent months, asking questions about a meeting that Mr. Trump held in February 2020 with officials who briefed him about election security for the upcoming race. The special counsel’s interest in the meeting, where Mr. Trump praised what officials told him were improvements in election security, was reported earlier by CNN.

During the meeting, Mr. Trump attacked Joseph Maguire, who was then serving as acting director of national intelligence, for having days earlier given a briefing on Russian interference in the 2016 election to Representative Adam Schiff, Democrat of California, then the chairman of the House Intelligence Committee, and other members of the panel, according to people familiar with the events.

Mr. Trump viewed Mr. Schiff as an enemy after he focused extensively on whether Mr. Trump’s campaign had conspired with Russia during his 2016 campaign and he played an instrumental role in his first impeachment.

At the meeting, officials from the F.B.I. and other agencies also told Mr. Trump about their preparations to secure the election from interference. Mr. Trump was so taken by what he heard that he wanted to hold a news conference to tout the security of the election, according to a person with knowledge of the talks.

Mr. Trump’s apparent excitement at the meeting could shed light on his state of mind and what factual knowledge he had as he spread baseless lies about election fraud months later.

In a related line of inquiry, prosecutors under Mr. Smith have asked questions as to when and how federal officials went about securing the election, and how they coordinated those efforts with secretaries of state in various states, according to a person familiar with the matter. Prosecutors have also sought to determine how regularly the White House was briefed on election security measures.

Richard Fausset contributed reporting.